Problem 20 - Precalculus Mathematics for Calculus

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$6-x \geq 2 x+9$$

Step-by-Step Solution

Answer

The solution is \(x \leq -1\), or \((-\infty, -1]\) in interval notation.

Step 1: Move Variable Terms to One Side

We start by moving all terms with the variable to one side of the inequality. We'll add \(x\) to both sides to isolate the variable terms on one side: \[ 6 - x + x \geq 2x + x + 9 \]This simplifies to: \[ 6 \geq 3x + 9 \]

Step 2: Isolate the Variable Term

Next, we need to isolate the \( x \) term. We do this by subtracting 9 from both sides:\[ 6 - 9 \geq 3x \]Simplifying this gives:\[ -3 \geq 3x \]

Step 3: Solve for the Variable

Now, solve for \(x\) by dividing both sides by 3:\[ \frac{-3}{3} \geq \frac{3x}{3} \]This simplifies to:\[ -1 \geq x \] or equivalently,\[ x \leq -1 \]

Step 4: Express Solution in Interval Notation

The solution \( x \leq -1 \) in interval notation is:\[ (-\infty, -1] \]This represents all values of \(x\) that are less than or equal to \(-1\).

Step 5: Graph the Solution Set

To graph the solution set, draw a number line. Place a closed circle (indicating \(\leq\)) at \(-1\) and shade the line to the left, representing all numbers less than or equal to \(-1\).

Key Concepts

Inequality Solving

Solving inequalities is similar to solving equations but requires a bit more attention. Inequalities are like equations but use symbols like ">", "<", "≥", or "≤" to show that one side is not exactly equal to the other.
The goal is to isolate the variable on one side of the inequality, just like with equations.

• First, you often need to simplify the inequality. This can involve combining like terms or getting all variable terms on one side. For our example, we added the same variable term on both sides.
• Next, you perform operations to solve for the variable. This could be adding or subtracting numbers or even dividing or multiplying to get the variable alone. Be very careful: if you multiply or divide by a negative number, you must flip the inequality symbol!

By following these steps on the example given, we moved all terms involving \(x\) to one side and solved to find \(x \leq -1\). This means any number less than or equal to \(-1\) satisfies the inequality.

Interval Notation

Interval notation is a concise way of writing a set of numbers, often used in conjunction with inequalities. It provides a simple and efficient way to express your solution set.

• The brackets or parentheses indicate whether endpoints are included or excluded. A square bracket "\([\)" or "\(]\)" means the endpoint is included in the solution set, while a parenthesis "\(()\)" means it is not.
• For example, the interval notation \(( -\infty, -1 ]\) means that the solution includes all numbers from negative infinity up to and including \(-1\). An infinity sign always uses a parenthesis because infinity itself is not a real number and can't be "inclusive."

Interval notation is not only widely used in simple inequalities, but also in calculus and other higher math courses. It's essential to understand how to read and write interval notation as it forms the basis of more complex math concepts.

Graphing Inequalities

Graphing inequalities visually represents the solution set on a number line, making it easier to understand which values satisfy the inequality.
Here's how to graph the solution \( x \leq -1 \):

• Start by drawing a horizontal number line. This line should have numbers evenly spaced, covering quite a bit above and below your possible solution.
• Locate the point at \(-1\) on the line. Since \(-1\) is part of the solution (as indicated by the inequality symbol \(\leq\)), place a closed circle (or dot) exactly on \(-1\).
• Shade the line to the left of this point. This shading represents every number less than \(-1\), which our solution accepts. Remember, shading to the left or right depends on whether your inequality is "less than" or "greater than." "Less than" heads left, and "greater than" heads right.

Graphing techniques are valuable since they give a quick visual clue to anyone reading the problem about what numbers fit the solution set, especially when dealing with complex inequalities.